[B] Simplify def. of continuous_2arg

This hides the destructuring of the pair inside the projection maps.
This makes proofs of `continuous_2arg` a lot easier, because now the
continuity of the projections and continuity of composition can be used
easier than before.

Author: Columbus240

theories/Topology/ProductTopology.v

@@ -358,11 +358,11 @@ Variable f:point_set X -> point_set Y -> point_set Z.
 
 Definition continuous_2arg :=
   continuous (fun p:point_set X * point_set Y =>
-              let (x,y):=p in f x y)
+                f (fst p) (snd p))
   (X:=ProductTopology2 X Y).
 Definition continuous_at_2arg (x:point_set X) (y:point_set Y) :=
   continuous_at (fun p:point_set X * point_set Y =>
-                 let (x,y):=p in f x y)  (x, y)
+                 f (fst p) (snd p))  (x, y)
   (X:=ProductTopology2 X Y).
 
 Lemma continuous_2arg_func_continuous_everywhere:
@@ -398,9 +398,10 @@ Lemma continuous_composition_at_2arg:
   continuous_at (fun w:point_set W => f (g w) (h w)) w.
 Proof.
 intros.
+red in H.
 apply (continuous_composition_at
   (fun p:point_set (ProductTopology2 X Y) =>
-      let (x,y):=p in f x y)
+      f (fst p) (snd p))
   (fun w:point_set W => (g w, h w))); trivial.
 now apply product2_map_continuous_at.
 Qed.

theories/Topology/RFuncContinuity.v

@@ -81,6 +81,7 @@ apply continuous_at_neighborhood_basis with
   + destruct x0 as [x' y'],
              H1 as [[[] []]].
     unfold R_metric.
+    simpl.
     replace (x'+y' - (x+y)) with ((x'-x) + (y'-y)) by ring.
     apply Rle_lt_trans with (Rabs (x'-x) + Rabs(y'-y)).
     * apply Rabs_triang.
@@ -167,9 +168,8 @@ pose proof (RTop_metrization 0).
 apply continuous_at_neighborhood_basis with
   (metric_topology_neighborhood_basis R_metric 0).
 - apply open_neighborhood_basis_is_neighborhood_basis.
-  replace (0*0) with 0 by auto with real.
-  apply H.
-
+  simpl. rewrite Rmult_0_r.
+  assumption.
 - intros.
   destruct H0.
   exists (characteristic_function_to_ensemble
@@ -192,9 +192,8 @@ apply continuous_at_neighborhood_basis with
     destruct x as [x y].
     destruct H1 as [[] []].
     unfold R_metric in *.
-    replace (x*y-0) with (x*y) by auto with real.
-    replace (x-0) with x in H1 by auto with real.
-    replace (y-0) with y in H2 by auto with real.
+    simpl.
+    rewrite Rminus_0_r in *.
     rewrite Rabs_mult.
     replace r with (r*1) by auto with real.
     apply Rmult_le_0_lt_compat;